Singular Values for Products of Complex Ginibre Matrices with a Source: Hard Edge Limit and Phase Transition

Abstract

The singular values squared of the random matrix product Y= GrGr - 1… G1(G0+ A) , where each Gj is a rectangular standard complex Gaussian matrix while A is non-random, are shown to be a determinantal point process with the correlation kernel given by a double contour integral. When all but finitely many eigenvalues of A*A are equal to bN, the kernel is shown to admit a well-defined hard edge scaling, in which case a critical value is established and a phase transition phenomenon is observed. More specifically, the limiting kernel in the subcritical regime of 0 < b 1 with two distinct scaling rates. Similar results also hold true for the random matrix product TrTr - 1… T1(G0+ A) , with each ..